Weighted polynomial approximation for weights with slowly varying extremal density
type: Item
instance of: scholarly article
title: Weighted polynomial approximation for weights with slowly varying extremal density
zbMATH Open document ID: 0936.41008
author: Vilmos Totik
published in: Journal of Approximation Theory
publication date: 1999-11-11T00:00:00Z
review text: In this paper approximation of continuous functions \(f\) by weighted polynomials of the form \(w^nP_n\) on a closed subset \(\Sigma\) of the real line is investigated. The weight \(w\) is a continuous function on \(\Sigma\), not identically vanishing. The polynomials \(P_n\) have degree at most \(n\). Approximability is governed by the known theorem: There exists a closed set \(S\subset \Sigma\) such that a continuous function \(f\) on \(\Sigma\) is the uniform limit of a sequence \(w^nP_n\), as \(n\to \infty\) if and only if \(f\) vanishes on \(S\). With \(w\) one can associate a probability measure \(\mu_w\) as the unique minimum solution of a weighted energy integral. When \(\mu_w\) has a density \(d\mu_w\) which is continuous and positive near a point \(x_0\in\Sigma\), then \(x_0\notin S\). It is shown in the paper under review that the same is true if \(d\mu_w\) is slowly varying and bounded away from 0 near \(x_0\). This allows logarithmic singularities. As second main result the author shows that under mild additional assumptions, \(x_0\notin S\) holds even when the density \(d\mu_w\) has an isolated zero at \(x_0\) provided \(d\mu_w\) is slowly varying. These results are in contrast to the known results which ensure that polynomial zeros or polynomial singularities \(x_0\) where the density \(d\mu_w\) behaves like \(|x-x_0|^\lambda\) with \(\lambda>0\) or \(\lambda<0\) must lie in \(S\).
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 1346403
zbMATH Keywords: weighted polynomial approximation
zbMATH Keywords: slowly varying functions
zbMATH Keywords: weighted energy problem
reviewed by: Rudolf Wegmann
MaRDI profile type: Publication
full work available at URL: jath.1998.3319
OpenAlex ID: W2053511324
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DOI: 10.1006/JATH.1998.3319
description: scientific article; zbMATH DE number 1346403